A DATABASE OF NUMBER FIELDS
JOHN W. JONES AND DAVID P. ROBERTS
Abstract. We describe an online database of number fields which accompanies this paper. The database centers on complete lists of number fields with
prescribed invariants. Our description here focuses on summarizing tables and
connections to theoretical issues of current interest.
1. Introduction
A natural computational problem is to completely determine the set K(G, D)
of all degree n number fields K with a given Galois group G ⊆ Sn and a given
discriminant D. Many papers have solved instances of this problem, some relatively
early contributions being [Hun57, Poh82, BMO90, SPDyD94].
This paper describes our online database of number fields at
http://hobbes.la.asu.edu/NFDB/ .
This database gives many complete determinations of K(G, D) in small degrees n,
collecting previous results and going well beyond them. Our database complements
the Klüners-Malle online database [KM01], which covers more groups and signatures, but is not as focused on completeness results and the behavior of primes.
Like the Klüners-Malle database, our database is searchable and intralinked.
Section 2 explains in practical terms how one can use the database. Section 3
explains some of the internal workings of the database, including how it keeps track
of completeness. Section 4 presents tables summarizing the contents of the database
in degrees n ≤ 11, which is the setting of most of our completeness results. The
section also briefly indicates how fields are chosen for inclusion on the database and
describes connections with previous work.
The remaining sections each summarize an aspect of the database, and explain
how the tabulated fields shed some light on theoretical issues of current interest.
As a matter of terminology, we incorporate the signature of a field into our notion
of discriminant, considering the formal product D = −s |D| to be the discriminant
of a field with s complex places and absolute discriminant |D|.
Section 5 focuses on the complete list of all 11279 quintic fields with Galois
group G = S5 and discriminant of the form −s 2a 3b 5c 7d . The summarizing table
here shows that the distribution of discriminants conforms moderately well to the
mass heuristic of [Bha07]. Section 6 summarizes lists of fields for more nonsolvable
groups, but now with attention restricted to discriminants of the form −s pa q b with
p < q primes.
2010 Mathematics Subject Classification. 11R21, 11R32.
This work was partially supported by a grant from the Simons Foundation (#209472 to David
Roberts).
1
2
JOHN W. JONES AND DAVID P. ROBERTS
Sections 7 and 8 continue to pursue cases with D = −s pa q b , but now for octic
groups G of 2-power order. Section 7 treats the cases p > 2 and discusses connections to tame maximal nilpotent extensions as studied in [BE11, BP00]. Section 8
treats the case p = 2 and takes a first step towards understanding wild ramification
in some of the nilpotent extensions studied in [Koc02].
Sections 9 and 10 illustrate progress in the database on a large project initiated
in [JR07]. The project is to completely classify Galois number fields with root discriminant |D|1/n at most the Serre-Odlyzko constant Ω := 8πeγ ≈ 44.76. Upper
bounds on degrees coming from analysis of Dedekind zeta functions [Mar82, Odl90]
play a prominent role. The database gives many solvable fields satisfying the root
discriminant bound. In this paper for brevity we restrict attention to nonsolvable fields, where, among other interesting things, modular forms [Bos07], [Sch12]
sometimes point the way to explicit polynomials.
The database we are presenting here has its origin in posted versions of the
complete tables of our earlier work [JR99]. Other complete lists of fields were posted
sporadically in the next ten years, while most fields and the new interface are recent
additions. Results from the predecessors of the present database have occasionally
been used as ingredients of formal arguments, as in e.g. [HKS06, OT05, Dah08].
The more common use of our computational results has been to guide investigations
into number fields in a more general way. With our recent enhancements and
this accompanying paper, we aim to increase the usefulness of our work to the
mathematical community.
2. Using the database
A simple way to use the database is to request K(G, D), for a particular (G, D).
A related but more common way is to request the union of these sets for varying
G and/or D. Implicit throughout this paper and the database is that fields are
always considered up to isomorphism. As a very simple example, asking for quartic
fields with any Galois group G and discriminant D satisfying |D| ≤ 250 returns
Table 2.1.
Results below are proven complete
rd(K) grd(K) D
h G Polynomial
3.29
6.24 −2 32 131 1 D4 x4 − x3 − x2 + x + 1
3.34
3.34 −2 53
1 C4 x4 − x3 + x2 − x + 1
2 4 2
3.46
3.46 − 2 3
1 V 4 x4 − x2 + 1
2 3 1
1 D4 x4 − x3 + 2x + 1
3.71
6.03 − 3 7
2 2 2
3.87
3.87 − 3 5
1 V4 x4 − x3 + 2x2 + x + 1
2
1
3.89
15.13 − 229
1 S4 x4 − x + 1
Table 2.1. Results of a query for quartic fields with absolute
discriminant ≤ 250, sorted by root discriminant
In general, the monic polynomial f (x) ∈ Z[x] in the last column defines the
field of its line, via K = Q[x]/f (x). It is standardized by requiring the sum of the
absolute squares of its complex roots to be minimal, with ties broken according to
the conventions of Pari’s polredabs. Note that the database, like its local analog
A DATABASE OF NUMBER FIELDS
3
[JR06], is organized around non-Galois fields. However, on a given line, some of
the information refers to a Galois closure K g .
The Galois group G = Gal(K g /Q) is given by its common name, like in Table 2.1,
or its T -name as in [BM83, PAR13, GAP06] if it does not have a very widely
accepted common name. Information about the group—essential for intelligibility
in higher degrees—is obtainable by clicking on the group. For example, the database
reports 10T 42 as having structure A25 .4, hence order 602 4 = 14400; moreover, it is
isomorphic to 12T 278, 20T 457, and 20T 461.
Continuing to explain Table 2.1, the column D prints −s |D|, where s is the
number of complex places and |D| is given in factored form. This format treats the
infinite completion Q∞ = R on a parallel footing with the p-adic completions Qp .
If n ≤ 11, then clicking on any appearing prime p links into the local database of
[JR06], thereby giving a detailed description of the p-adic algebra Kp = Qp [x]/f (x).
This automatic p-adic analysis also often works in degrees n > 11.
The root discriminant rd(K) = |D|1/n is placed in the first column, since one
commonly wants to sort by root discriminant. Here and later we often round real
numbers to the nearest hundredth without further comment. When it is implemented, our complete analysis at all ramifying primes p automatically determines
the Galois root discriminant of K, meaning the root discriminant of a Galois closure K g . The second column gives this more subtle invariant grd(K). Clicking on
the entry gives the exact form and its source. Often it is better to sort by this
column, as fields with the same Galois closure are then put next to each other. As
an example, quartic fields with G = D4 come in twin pairs with the same Galois
closure. The twin K t of the first listed field K on Table 2.1 is off the table because
|D(K t )| = 31 132 = 504; however grd(K t ) = grd(K) = 31/2 131/2 ≈ 6.24.
Class numbers are given in the column h, factored as h1 · · · hd where the class
group is a product of cyclic groups of size hi . There is a toggle button, so that
one can alternatively receive narrow class numbers in the same format. To speed
up the construction of the table, class numbers were computing assuming the generalized Riemann hypothesis; they constitute the only part of the database that is
conditional. Theoretical facts about class groups can be seen repeatedly in various
parts of the database. For example, let n be an odd positive integer and consider
a degree n field K with dihedral Galois group Dn . Let L be its Galois closure
with unique quadratic subfield F . Let p be a prime not dividing 2n and consider
the p-parts of all class groups in question. Then, decomposing via the natural Dn
action on Clp (L) and using the triviality of Clp (Q), one gets
(2.1)
Clp (L) ∼
= Clp (K)2 × Clp (F ).
One explicit example comes from the unique D7 field K with Galois root discriminant 19871/2 ∼
= 44.58. Illustrating (2.1), the database reports Cl(L) = 13 · 13,
Cl(K) = 13, and Cl(F ) = 7.
When the response to a query is known to be complete, then the table is headed
by the completeness statement shown in Table 2.1. As emphasized in the introduction, keeping track of completeness is one of the most important features of
the database. The completeness statement often reflects a very long computational
proof, even if the table returned is very short.
There are many other ways to search the database, mostly connected to the
behavior of primes. For example, one can restrict the search to find fields with
4
JOHN W. JONES AND DAVID P. ROBERTS
restrictions on ordp (D), or one can search directly for fields with Galois root discriminant in a given range. On the other hand, there are some standard invariants
of fields that the database does not return, such as Frobenius partitions and regulators. The database does allow users to download the list of polynomials returned,
so that it can be used as a starting point for further investigation.
3. Internal structure
The website needs to be able to search and access a large amount of information.
It uses a fairly standard architecture: data is stored in a MySQL database and web
pages are generated by programs written in Perl.
A MySQL database consists of a collection of tables where each table is analogous
to a single spreadsheet with columns representing the types of data being stored. We
use data types for integers, floating point numbers, and strings, all of which come
in various sizes, i.e., amount of memory devoted to a single entry. When searching,
one can use equalities and inequalities where strings are ordered lexicographically.
When a user requests number fields, the Perl program takes the following steps:
(1) Construct and execute a MySQL query to pull fields from the database.
(2) Filter out fields which satisfy all of the user’s requirements when needed
(see below).
(3) Check completeness results known to the database.
(4) Generate the output web page.
The main MySQL table has one row for each field. There are columns for each
piece of information indicated by the input boxes in the top portion of the search
screen, plus columns for the defining polynomial (as a string), and an internal
identifier for the field. The only unusual aspect of this portion of the database
is how discriminants are stored and searched. The difficulty stems from the fact
that many number fields in the database have discriminants which are too large
to store in MySQL as integers. An option would be to store the discriminants
as strings, but then it would be difficult to search for ranges: string comparisons
in MySQL are lexicographic, so ‘11’ comes before ‘4’. Our solution is to store
absolute discriminants |D| as strings, but prepend the string with four digits which
give blog10 (|D|)c, padded on the left with zeros as needed. So, 4 is stored as ‘00004’,
11 is ‘000111’, etc. This way we can use strings to store each discriminant in its
entirety, but searches for ranges work correctly.
The MySQL table of number fields also has a column for the list of all primes
which ramify in the field, stored as a string with a separator between primes. This
is used to accelerate searches when it is clear from the search criteria that only a
small finite list of possibilities can occur, for example, when the user has checked
the box that “Only listed primes can ramify”.
Information on ramification of specific primes can be input in the bottom half
of the search inputs. To aid in searches involving these inputs, we have a second MySQL table, the ramification table, which stores a list of triples. A triple
(field identifier, p, e) indicates that pe exactly divides the discriminant of the corresponding field. The most common inputs to the bottom half of the search page
work well with this table, namely those which list specific primes and allowable discriminant exponents. However, the search boxes allow much more general inputs,
i.e., where a range of values is allowed for the prime and the discriminant exponent
allows both 0 and positive values. It is possible to construct MySQL queries for
A DATABASE OF NUMBER FIELDS
5
inputs of this sort, but they are complicated, involve subqueries, and are relatively
slow. Moreover, a search condition of this type typically rules out relatively few
number fields. If a user does make such a query, we do not use the information
at this stage. Instead, we invoke Step (2) above to select fields from the MySQL
query which satisfy these additional requirements.
The database supports a variety of different types of completeness results. Complicating matters is that these results can be interrelated. We use four MySQL
tables for storing ways in which the data is complete. In describing them, G denotes the Galois group of a field, n is the degree, s is the number of complex places,
and |D| is the absolute discriminant, as above. The tables are
A. store (n, s, B) to indicate that the database is complete for fields with the
given n and s such that |D| ≤ B;
B. store (n, s, G, B) to indicate that the database is complete for fields with
the given n, s, and G such that |D| ≤ B;
C. store (n, S, L) where S is a list of primes and L is a list of Galois groups to
indicate that the database is complete for degree n fields unramified outside
S for each Galois group in L;
D. store (n, G, B) to indicate that the database is complete for degree n fields
K with Galois group G such that grd(K) ≤ B.
In each case, database entries include the degree, so individual Galois groups can
be stored by their T -number (a small integer). In the third case, we store the list
L by an integer whose bits indicate which T -numbers are included in the set. For
example, there are 50 T -numbers in degree 8, so a list of Galois groups
P in that
degree is a subset of S ⊆ {1, . . . , 50} which we represent by the integer t∈S 2t−1 .
These integers are too large to store in the database as integers, so they are stored
as strings, and converted to multiprecision integers in Perl. The list of primes in
the third table is simply stored as a string consisting of the primes and separating
characters.
To start checking for completeness, we first check that there are only finitely
many degrees involved, and that the search request contains an upper bound on
at least one of: |D|, rd(K), grd(K), or the largest ramifying prime. We then loop
over the degrees in the user’s search. We allow for the possibility that a search is
known to be complete by some combination of completeness criteria. So throughout
the check, we maintain a list of Galois groups which need to be checked, and the
discriminant values to check. If one check shows that some of the Galois groups
for the search are known to be complete, they are removed from the list. If that
list drops to being empty, then the search in that degree is known to be complete.
Discriminant values are treated analogously.
For each degree, bounds on |D| and rd(K) are clearly equivalent. Less obviously,
bounds between rd(K) and grd(K) are related. In particular, we always have
rd(K) ≤ grd(K), but also have for each Galois group, grd(K) ≤ rd(K)α(G) where
α(G) is a rational number depending only on G (see [JRb]).
We then perform the following checks.
• We compare the request with Tables A, B, and D for discriminant bound
restrictions.
• Remove Galois groups from the list to be checked based on grd.
• If there are at most ten discriminants not accounted for, check each individually against Table C.
6
JOHN W. JONES AND DAVID P. ROBERTS
• If there is a bound on the set of ramifying primes, which could arise from
the user checking “Only these primes ramify”, or from a bound on the
maximum ramifying prime, check Table C.
4. Summarizing tables
The tables of this section summarize all fields in the database of degree ≤ 11.
Numbers on tables which are known to be correct are given in regular type. Numbers which are merely the bounds which come from perhaps incomplete lists of
fields are given in italics. The table has a line for each group nT j, sorted by degree
n and the index j. A more descriptive name is given in the second column.
T
1
G
2
{2, 3}
7
{2, 5}
7
{3, 5}
3
Degree 2
{2, 3, 5} rd(K)
15
1.73
grd(K)
1.73
|K[G, Ω]|
1220
Tot
1216009
T
1
2
G
3
S3
{2, 3}
1
8
{2, 5}
0
1
{3, 5}
1
5
Degree 3
{2, 3, 5} rd(K)
1
3.66
31
2.84
grd(K)
3.66
4.80
|K[G, Ω]|
47
610
Tot
1004
856373
T
1
2
3
4
5
G
4
22
D4
A4
S4
{2, 3}
4
7
28
1
22
{2, 5}
12
7
24
0
3
{3, 5}
2
1
0
0
1
Degree 4
{2, 3, 5} rd(K)
24
3.34
35
3.46
176
3.29
1
7.48
143
3.89
grd(K)
3.34
3.46
6.03
10.35
13.56
|K[G, Ω]|
228
2421
2850
59
527
Tot
9950
52469
764341
28786
281089
T
1
2
3
4
5
G
5
D5
F5
A5
S5
{2, 3}
0
0
1
0
5
{2, 5}
1
4
19
5
38
{3, 5}
1
2
7
6
22
Degree 5
{2, 3, 5} rd(K)
1
6.81
8
4.66
82
8.11
62
7.14
1353
4.38
grd(K)
6.81
6.86
11.08
18.70
24.18
|K[G, Ω]|
7
146
102
78
192
Tot
181
11516
1645
95337
598542
The next four columns represent a main focus of the database, complete lists
of fields ramified within a given set of primes. As a matter of notation, we write
e.g. K(G, −∗ p∗ q ∗ ) to denote the union of all K(G, −s pa q b ). The database contains
completeness results for many other prime combinations beyond those given in the
table; §5-§8 give examples of these further completeness results.
The next column gives minimal values of root discriminants. More refined minima can easily be obtained from the database. For example, for S5 , minimal discriminants for s = 0, 1, and 2 complex places are respectively (61 · 131)1/5 ≈ 7.53,
(13 · 347)1/5 ≈ 5.38, and 16091/5 ≈ 4.38. Completeness is typically known well past
the minimum.
In understanding root discriminants, the Serre-Odlyzko constant Ω = 8πeγ ≈
44.76 mentioned in the introduction plays an important role as follows. First, if K
A DATABASE OF NUMBER FIELDS
7
Degree 6
T
G
{2, 3} {2, 5} {3, 5} {2, 3, 5} rd(K) grd(K) |K[G, Ω]|
Tot
1
6
7
0
3
15
5.06
5.06
399
5291
2
S3
8
1
5
31
4.80
4.80
610
8353
3
D6
48
6
10
434
4.93
8.06
3590 147965
4
A4
1
0
0
1
7.32
10.35
59
1357
5
3o2
8
0
5
31
4.62
10.06
254
2169
6
2o3
7
0
0
15
5.61
12.31
243 62484
7
S4+
22
3
1
143
5.69
13.56
527 242007
8
S4
22
3
1
143
6.63
13.56
527 18738
9
S32
22
0
4
375
7.89
15.53
445
9721
10
32 : 4
4
0
2
44
8.98
23.57
34
396
11
2 o S3
132
18
2
2002
4.65
16.13
2196 323148
12 P SL2 (5)
0
5
6
62
8.12
18.70
78
275
13 32 : D4
50
0
0
624
4.76
21.76
274 27049
14 P GL2 (5)
5
38
22
1353 11.01
24.18
192 11519
15
A6
8
2
4
540
8.12
31.66
10
670
16
S6
54
30
42
8334
4.95
33.50
26 21594
T
1
2
3
4
5
6
7
G
7
D7
7:3
7:6
SL3 (2)
A7
S7
Degree 7
{2, 3} {2, 5} {3, 5} {2, 3, 5} rd(K) grd(K) |K[G, Ω]|
0
0
0
0 17.93
17.93
4
0
0
0
0
6.21
8.43
80
0
0
0
0 21.03
31.64
2
0
0
1
5 12.10
15.99
94
0
0
0
7.95
32.25
36
0
2
3
204
8.74
39.52
1
10
24
14
4391
5.65
40.49
1
Tot
117
496
56
189
618
331
8357
has root discriminant < Ω/2, then its maximal unramified extension K 0 has finite
degree over K. Second, if rd(K) < Ω, then the generalized Riemann hypothesis
implies the same conclusion [K 0 : K] < ∞. Third, suggesting that there is a
modestly sharp qualitative transition associated with Ω, the field Q(e2πi/81 ) with
root discriminant 33.5 ≈ 46.77 has [K 0 : K] = ∞ by [Hoe09].
The next two columns of the tables again represent a main focus of the database,
complete lists of fields with small Galois root discriminant. We write K[G, B] for
the set of all fields with Galois group G and grd(K) ≤ B. The tables give first the
minimal Galois root discriminant. They next give |K[G, Ω]|. For many groups, the
database is complete for cutoffs well past Ω. For example, the set K[9T 17, Ω] is
empty, and not adequate for the purposes of [JRa]. However the database identifies
|K[9T 17, 200]| = 36 and this result is adequate for the application.
The last column in a table gives the total number of fields in the database for the
given group. Note that one could easily make this number much larger in any case.
For example, a regular family over Q(t) for each group is given in [MM99, App. 1],
and one could simply specialize at many rational numbers t. However we do not
do this: all the fields on our database are there only because discriminants met one
criterion or another for being small. The fluctuations in this column should not be
8
JOHN W. JONES AND DAVID P. ROBERTS
Degree 8
T
G
{2, 3} {2, 5} {3, 5} {2, 3, 5} rd(K) grd(K) |K[G, Ω]|
Tot
1
8
4
8
0
16 11.93
11.93
23
5777
2
4×2
6
18
1
84
5.79
5.79
581 15412
3
23
1
1
0
15
6.93
6.93
908 10687
4
D4
14
12
0
88
6.03
6.03
1425 24370
5
Q8
2
0
0
8 18.24
18.24
7
778
6
D8
20
20
0
104
6.71
9.75
708 29740
7
8 : {1, 5}
6
20
1
88
9.32
9.32
55
8040
8
8 : {1, 3}
22
10
0
120 10.09
10.46
121 10826
9
D4 × 2
28
24
0
528
6.51
10.58
5908 175572
10
22 : 4
8
24
0
160
6.09
9.46
620 29649
11
Q8 : 2
18
18
0
312
6.51
9.80
921 17350
12
SL2 (3)
0
0
0
0 12.77
29.84
4
681
13
A4 × 2
7
0
0
15
8.06
12.31
243 26637
14
S4
22
3
1
143
9.40
13.56
527
7203
15
8 : 8×
42
42
0
928
8.65
13.79
818 60490
16
1/2[24 ]4
8
24
0
176
7.45
13.56
76 15545
17
4o2
16
72
0
480
5.79
13.37
252 42018
18
22 o 2
24
8
0
608
7.04
16.40
2544 216411
19
E(8) : 4
8
24
0
192
9.51
14.05
220 24380
20
[23 ]4
4
12
0
96
8.46
14.05
110 13631
21 1/2[24 ]E(4)
4
12
0
96
8.72
14.05
110 10091
22 E(8) : D4
0
0
0
204
8.43
18.42
882 19733
23
GL2 (3)
128
24
4
912
8.31
16.52
388
6304
24
S4 × 2
132
18
2
2002
6.04
16.13
2196 45996
25
23 : 7
0
0
0
0 12.50
17.93
1
20
26 1/2[24 ]eD(4)
64
24
0
1872
7.23
20.37
840 135840
27
2o4
16
48
0
448
5.95
19.44
160 86501
28 1/2[24 ]dD(4)
16
48
0
448
8.67
19.44
160 47150
29 E(8) : D8
48
24
0
1296
6.58
19.41
1374 170694
30 1/2[24 ]cD(4)
16
48
0
448
8.25
19.44
140 48249
31
2 o 22
16
8
0
432
5.92
19.41
458 54843
32
[23 ]A4
0
0
0
0 13.56
34.97
24 29970
33
E(8) : A4
6
0
0
14 13.73
30.01
24
3240
34 E(4)2 : D6
11
1
0
132 14.16
27.28
55
3907
35
2 o D(4)
168
72
0
5568
5.83
22.91
1464 729730
36
23 : 7 : 3
0
0
0
0 14.37
31.64
4
298
37
P SL2 (7)
0
0
21.00
32.25
18
352
38
2 o A4
24
0
0
112 10.66
37.27
46 67160
39
[23 ]S4
168
24
0
2496
6.73
32.35
84 24625
40 1/2[24 ]S(4)
216
24
0
3872
7.67
29.71
98 12796
41
E(8) : S4
90
12
0
2282
8.38
28.11
222 11929
42
A4 o 2
12
0
0
83
7.68
32.18
14
3432
43
P GL2 (7)
4
8 11.96
27.35
27
1495
44
2 o S4
656
96
0
22944
5.84
31.38
336 440683
45
[1/2.S42 ]2
110
0
0
836
9.28
29.35
39
7732
46 1/2[S(4)2 ]2
28
0
0
54 11.35
49.75
0
216
47
S4 o 2
542
0
0
2185
6.74
35.05
15 28765
48 23 : SL3 (2)
0
0
11.36
39.54
6
495
49
A8
2
4
1
55 15.24
72.03
90
50
S8
72
30
9
1728 11.33
43.99
1
4026
A DATABASE OF NUMBER FIELDS
Degree 9
T
G
{2, 3} {2, 5} {3, 5} {2, 3, 5}
1
9
1
0
1
1
2
32
0
0
0
0
3
D9
6
0
4
20
4
S3 × 3
8
0
5
31
5
32 : 2
1
0
1
15
6
1/3[33 ]3
0
0
0
0
7
32 : 3
0
0
0
0
8
S3 × S3
22
0
4
375
9
E(9) : 4
2
0
1
22
10
[32 ]S(3)6
22
0
17
171
11
E(9) : 6
6
0
4
20
12
[32 ]S(3)
12
0
12
180
13
E(9) : D6
6
0
4
20
14
32 : Q8
4
0
0
19
15
E(9) : 8
5
1
0
18
16
E(9) : D8
25
0
0
312
17
3o3
0
0
0
0
18
E(9) : D12
80
0
8
1380
19
E(9) : 2D8
60
1
0
124
20
3 o S3
18
0
12
60
21 1/2.[33 : 2]S3
54
0
54
1296
22
[33 : 2]3
18
0
12
60
23
E(9) : 2A4
0
0
0
0
24
[33 : 2]S(3)
321
0
48
8307
25 [1/2.S(3)3 ]3
4
0
0
4
26
E(9) : 2S4
250
2
10
362
27
P SL2 (8)
4
4
28
S3 o 3
28
0
0
90
29 [1/2.S(3)3 ]S(3)
45
0
1
512
30 1/2[S(3)3 ]S(3)
232
1
40
1637
31
S3 o S3
616
0
5
19865
32
ΣL2 (8)
64
240
33
A9
13
2
314
34
S9
46
1
1
1507
9
rd(K) grd(K) |K[G, Ω]|
Tot
13.70
13.70
3
52
15.83
15.83
9
189
9.72
12.19
105
705
8.38
10.06
254 10139
14.29
15.19
48
373
17.63
31.18
2
85
26.09
50.20
0
90
8.93
15.53
445
7055
19.92
23.57
17
142
9.57
17.01
69
1066
14.67
16.83
64
880
8.92
16.72
148 13929
10.98
16.83
64
642
21.52
29.72
2
47
17.74
25.41
3
40
9.19
21.76
137
434
14.93
75.41
0
1274
8.53
22.06
290
9260
17.89
23.41
33
624
7.83
29.89
30
7989
9.82
24.90
126
4282
10.27
26.46
51
784
16.48
49.57
0
40
9.15
30.64
111 17973
12.89
29.96
4
303
12.79
27.88
51
866
16.25 30.31
15
19
8.18
33.56
7
6738
9.38
40.81
2
1255
6.86
30.37
35
5026
6.83
36.26
15 112887
16.09 34.36
15
1141
14.17 62.12
627
7.84 53.19
3189
viewed as significant, as the criteria depend on the group in ways driven erratically
by applications.
There are a number of patterns on the summarizing tables which hold because
of relations between transitive groups. For example the groups 5T 4 = A5 , 6T 12 =
P SL2 (5), and 10T 7 are all isomorphic. Most of the corresponding lines necessarily
agree. Similarly A5 is a quotient of 10T 11, 10T 34, and 10T 36. Thus the fact
that K(A5 , −∗ 2∗ 3∗ ) = ∅ immediately implies that also K(10T j, −∗ 2∗ 3∗ ) = ∅ for
j ∈ {11, 34, 36}.
Almost all fields in the database come from complete searches of number fields
carried out by the authors. In a few cases, we obtained polynomials from other
sources, notably for number fields of small discriminant: those compiled by the
Bordeaux group [Bor], which in turn were computed by several authors, and the
10
JOHN W. JONES AND DAVID P. ROBERTS
T
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
Degree 10
G
{2, 3} {2, 5} {3, 5} {2, 3, 5} rd(K) grd(K) |K[G, Ω]|
Tot
10
0
7
3
15
8.65
8.65
69
360
D5
0
4
2
8
6.86
6.86
146
822
D10
0
24
4
112
8.08
10.91
768
857
1/2[F (5)]2
1
19
7
82 10.23
11.08
102
178
F5 × 2
6
114
14
1148
9.48
14.50
584
1611
[52 ]2
0
8
4
16
6.84
18.02
32
175
A5
0
5
6
62 12.35
18.70
78
146
[24 ]5
0
3
0
3 12.75
24.98
18
36
[1/2.D(5)2 ]2
0
12
2
56 12.71
24.72
34
87
1/2[D(5)2 ]2
0
22
12
126 14.02
24.00
22
144
A5 × 2
0
35
18
930
9.42
22.24
179
1177
S5 (10a)
5
38
22
1353 12.04
24.18
192
1560
S5 (10d)
5
38
22
1353
9.16
24.18
192
1712
2o5
0
21
0
45
9.32
26.08
45
2050
[24 ]D(5)
0
60
0
360
9.33
25.15
72
620
1/2[25 ]D(5)
0
60
0
360
9.46
25.15
72
509
[52 : 4]2
0
59
0
916 17.46
26.65
34
1094
[52 : 4]22
0
16
0
17 19.75
35.98
3
22
[52 : 42 ]2
0
17
0
63 16.96
28.08
18
111
[52 : 42 ]22
0
0
0
31 27.36
48.25
0
43
D5 o 2
0
34
0
118
7.54
28.08
36
235
S5 × 2
30
228
44
18942
7.06
26.99
570
26851
2 o D5
0
360
0
5040
7.26
26.26
240
24024
[24 ]F (5)
7
173
0
1250 14.13
27.62
30
1491
1/2[25 ]F (5)
7
173
0
1250 13.84
27.62
30
1491
P SL2 (9)
4
1
2
270 20.20
31.66
5
334
[1/2.F52 ]2
0
56
0
652 13.40
40.43
18
1052
1/2[F52 ]2
1
24
2
57 15.16
32.71
8
59
2 o F5
42 1038
0
17500 11.44
32.17
90
19112
P GL2 (9)
11
5
1
55 22.64
34.42
6
149
M10
20
4
13
83 27.73
53.50
198
S6
27
15
21
4166 14.74
33.50
13
6913
F5 o 2
0
177
0
484
9.93
35.41
3
485
[24 ]A5
0
35
0
1322 10.82
35.81
5
1388
P ΓL2 (9)
100
32
15
1666 17.98
38.61
15
3531
2 o A5
0
245
0
19830 10.39
36.60
8
20660
[24 ]S5
91
450
8
42059
7.80
38.11
17
60029
1/2[25 ]S5
91
450
8
42059
9.41
38.11
17
42851
2 o S5
546 2700
16 588826
6.79
38.11
30 1095840
A5 o 2
12
63
29
1093
9.48
41.90
1
1124
[A5 : 2]2
28
139
11
9435
9.30
43.89
1
9677
1/2[S52 ]2
18
68
31
850 14.35
45.93
882
S5 o 2
185
422
20743
6.82
48.97
31847
A10
23
16
6
801 19.37
51.68
1201
S10
1
12
3
2585
7.77
70.36
4944
the tables of totally real fields of Voight [Voi08, Voi]. In addition, we include fields
found by the authors in joint work with others [DJ10, JW12].
A DATABASE OF NUMBER FIELDS
11
Degree 11
T
G
{2, 3} {2, 5} {3, 5} {2, 3, 5} rd(K) grd(K) |K[G, Ω]| Tot
1
11
0
0
0
0 17.30
17.30
1
18
2
D11
0
0
0
0 10.24
12.92
32
55
3
11 : 5
0
0
0
0 88.82 105.74
0
2
4
11 : 10
17.01
20.70
4
55
5 P SL2 (11)
15.36
42.79
2
91
6
M11
1 96.24 270.83
10
7
A11
4 21.15 146.24
71
8
S11
5
4
1
123
7.72
91.50
931
To compute cubic fields, we used the program of Belabas [Bel04, Bel]. Otherwise,
we obtain complete lists by using traditional and targeted Hunter searches [JR99,
JR03] or the class field theory functions in pari/gp [PAR13]. For larger nonsolvable
groups where completeness results are currently out of reach, we obtain most of our
fields by specializations of families at carefully chosen points to keep ramification
small in various senses.
5. S5 quintics with discriminant −s 2a 3b 5c 7d
One of our longest searches determined K(S5 , −∗ 2∗ 3∗ 5∗ 7∗ ), finding it to consist of
11279 fields. In this section, we consider how this set interacts with mass heuristics.
In general, mass heuristics [Bha07, Mal02] give one expectations as to the sizes
|K(G, D)| of the sets contained in the database. Here we consider these heuristics
only in the most studied case G = Sn . The mass of a Qv -algebra Kv is by definition
1/|Aut(Kv )|. Thus the mass of Rn−2s Cs is
1
(5.1)
µn,−s =
.
(n − 2s)!s!2s
For p a prime, similarly let µn,pc be the total mass of all p-adic algebras with degree
n and discriminant pc . For n < p, all algebras involved are tame and
(5.2)
µn,pc = |{Partitions of n having n − c parts}|.
For n ≥ p, wild algebras are involved. General formulas for µn,pc are given in
[Rob07].
Q
The mass heuristic says that if the discriminant D = −s p pcp in question is a
non-square, then
Y
Y
(5.3)
|K(Sn , −s
pcp )| ≈ δn µn,−s
µn,pcp .
p
p
Here δn = 1/2, except for the special cases δ1 = δ2 = 1 which require adjustment
for simple reasons. The left side is an integer and the right side is often close to
zero because of (5.1) and (5.2). So (5.3) is intended only to be used in suitable
averages.
For n ≤ 5 fixed and |D| → ∞, the heuristics are exactly right on average, the
case n = 3 being the Davenport-Heilbronn theorem and the cases n = 4, 5 more
recent results of Bhargava [Bha05, Bha10]. For a fixed set of ramifying primes S
and n → ∞, the mass heuristic predicts no fields after a fairly sharp cutoff N (S),
while in fact there can be many fields in degrees well past this cutoff [Rob]. Thus
the regime of applicability of the mass heuristic is not clear.
12
JOHN W. JONES AND DAVID P. ROBERTS
To get a better understanding of this regime, it is of interest to consider other
limits. Let cn be the number of elements of order ≤ 2 in Sn . Let µn,p∗ be the total
mass of all Qp -algebras of degree n. Thus µn,p∗ is the number of partitions of n if
n < p. Then, for k → ∞, the mass heuristic predicts the asymptotic equivalence
∗ ∗
|K(Sn , − 2
(5.4)
· · · p∗k })|
∼ δn µ
n,−∗
k
Y
µn,p∗ .
j=1
Both sides of (5.4) are 1 for all k when n = 1. For n = 2 and k ≥ 1, the statement
becomes 2k+1 −1 ∼ 2k+1 which is true. Using the fields in the database as a starting
point, we have carried out substantial calculations suggesting that, after removing
fields with discriminants of the form −3u2 from the count on the left, (5.4) holds
also for n = 3 and n = 4.
In this section, we focus on the first nonsolvable case, n = 5. For k ≥ 3, (5.4)
becomes
1 26
· 40 · 19 · 27 · 7k−3 ≈ 6.48 · 7k .
(5.5)
|K(S5 , −∗ 2∗ · · · p∗k )| ∼ ·
2 120
Through the cutoff k = 4, there are fewer fields than predicted by the mass heuristic:
pk
|K(S5 , −∗ 2∗ · · · p∗k )|
2
0
0%
3
5
6%
5
1353
61%
7
11279 .
72%
For comparison, the ratio 11279/(6.48 · 74 ) ≈ 72% is actually larger than the ratios
at k = 4 for cubic and quartic fields with discriminants −3u2 removed, these being
respectively 64/(1.33 · 34 ) ≈ 47% and 740/(3.30 · 5k ) ≈ 36%. As remarked above,
these other cases experimentally approach 100% as k increases. This experimental
finding lets one reasonably argue that (5.5) may hold too, with the small percentage
72% being a consequence of a small discriminant effect.
v\c 0
∞ 1
0.71
2 1
0.73
3 1
0.76
5 1
0.37
7 1
0.84
1
2
10 15
9.52 15.77
2
1.66
1
1
0.85 0.78
1
2
0.39 0.96
1
2
0.88 1.92
3
4
5
6
7
2
1.48
3
2.89
2
1.32
2
2.12
5
4.71
5
5.24
4
3.83
5
5.13
4
4.07
6
5.66
3
3.43
4
4
4.17 4.70
8
9
10
11
4
4
4
8
4.47 4.37 4.15 8.94
Total
26
40
19
4
5
4.65 6.38
1
1.24
27
7
Table 5.1. Local masses 120µ5,−c and µ5,pc , compared with
frequencies of local discriminants from K(S5 , −∗ 2∗ 3∗ 5∗ 7∗ ).
Table 5.1 compares local masses with frequencies of actually occurring local discriminants, inflated by the ratio (6.58 · 74 )/11279 to facilitate direct comparison.
Thus, e.g., the 7-adic discriminants (70 , 71 , 72 , 73 , 74 ) are predicted by the mass
heuristic to occur with relatively frequency (1, 1, 2, 2, 1). They actually occur with
A DATABASE OF NUMBER FIELDS
13
relative frequency (0.84, 0.88, 1.92, 2.12, 1.24). Here and for the other four places,
trends away from the predicted asymptotic values are explained by consistent underrepresentation of fields with small discriminant. The consistency of the data
with the mass heuristic on this refined level provides further support for (5.5).
6. Low degree nonsolvable fields with discriminant −s pa q b
Our earliest contributions to the general subject of number field tabulation were
[JR99] and [JR03]. These papers respectively found that there are exactly 398
sextic and 10 septic fields with discriminant of the form −s 2a 3b . On the lists from
these papers, the nonsolvable groups P SL2 (5) ∼
= A6 , A6 , S6 and
= A5 , P GL2 (5) ∼
SL3 (2), A7 , S7 respectively arise 0, 5, 8, 54, and 0, 0, 10 times. In this section, we
summarize further results from the database of this form, identifying or providing
lower bounds for |K(G, −∗ p∗ q ∗ )|.
2 3 5
2 • 5 38
3
• 22
5 5 6 •
7
11
2
13
1
17 1 1
19 1 3
23
1
29 2 3
31 1 3 1
37
1
41 2 2 2
43 1 3 1
47
2
53
59 1 3 2
61 1 1
67 2 1 1
71 1 2 1
73 1
1
79 4 4 2
83
1
89 1 3 2
97
1
T 24 40 28
7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 T
2 2 4 3 2 5 6 3 6 9 14 10 11 8 13 13 8 5 11 10 13 4 205
1 4
1 2 2 6 3 5 3 3 2 8 3 2 4 3 2 81
4 5 9 12 8 8 8 9 13 12 11 8 14 8 15 13 14 9 14 11 11 14 290
• 1
1 2
2 2
1 3
1
1
20
•
1 1
1
1 1
1
1 1
1 1
18
•
1
2
3 2
1
1
1 1 25
• 1
3
3 4 1
1 1
1 2
1 3 37
• 2
3
1
1 1 1 1 3 2
1 1
2 36
• 1
1
2
1 1
2 1 2 28
1 2
2 •
2 3 1 1
2 3 3 1 4 2
2 2 48
2 1
• 1
4 1
2 2 1
2
1 30
• 2 3
2 2 2
3 4 1 1 52
1 1 1
• 1
2
3
1 1 2 1 3 5 56
1
1 •
2 1 1 2 2 4 1 2
61
• 1 4
3
1 3
38
1
1
• 3 3
1 2 1 3 2
52
1 2
1
•
1 1 2 1 3 1 45
1
1
1 1
• 2 3 2 1
1 56
1
• 4
2 2 1 2 54
1
1 4 2
2 1
•
4 1 3
59
1
1
2 1 1
1
•
2 3 38
1
2
3
1 1
•
2 1 54
1
1
2 • 4
51
1 1 1 2
2
1 1 1
2
• 1 58
1
1
1
1
1
1 • 46
1 4 7 9 8 5 14 14 5 19 8 2 8 14 9 8 16 9 23 5 19 7
Table 6.1.
|K(A5 , −∗ p∗ q ∗ )| beneath the diagonal and
∗ ∗ ∗
|K(S5 , − p q )| above the diagonal. It is expected that A5 totals are smaller for primes p ≡ 2, 3 (5) because in this case p4 is
not a possible local discriminant.
14
JOHN W. JONES AND DAVID P. ROBERTS
The format of our tables exploits the fact that in the range considered for a given
group, there are no fields ramified at one prime only. In fact [JR08], the smallest
prime p for which K(G, −∗ p∗ ) is nonempty is as follows:
G
p
A5
653
S5
A6
101 1579
S6 SL3 (2) A7
197
227 > 227
S7
191
P GL2 (7)
.
53
Here 7T 5 = SL3 (2) is abstractly isomorphic to 8T 37 = P SL2 (7) and thus has
index two in 8T 43 = P GL2 (7).
2
3
5
2 • 54 30
3 8
• 42
5 2
4
•
7
2
11
13
2
17
2
19
2
2
23 2
29
4
31 4
6
T 16 30
8
7
2
•
11
4
2
13 17 19 23 29 31
2 2 2 4 4 6
8
2 12 2
2
6 8 2 4
•
•
•
2
•
2
•
2
2
4
2
6
4
•
2 •
8 12
T
104
124
98
2
6
4
12
8
16
18
12
Table 6.2.
|K(A6 , −∗ p∗ q ∗ )| beneath the diagonal and
∗ ∗ ∗
|K(S6 , − p q )| above the diagonal. All entries are even because
contributing fields come in twin pairs.
2
3
5
7
11
13
SL3 (2) and P GL2 (7)
2
3 5
7 11 13
•
4 0 51 0 0
0
• 0 28 0 0
0
0 •
4 0 0
44 12 4
• 4 6
4
0 0
6
• 0
0
0 0
0
0 •
2
2 •
3 0
5 2
7 0
11 0
13 0
A7
3
10
•
3
7
0
0
and
5
24
14
•
5
1
0
S7
7
55
44
18
•
0
0
11
0
2
0
5
•
0
13
0
0
0
0
0
•
Table 6.3. Determinations or lower bounds for |K(G, −∗ p∗ q ∗ )|
for four G. The entries |K(SL3 (2), −∗ p∗ q ∗ )| are all even because
contributing fields come in twin pairs.
Restricting to the six groups G of the form An or Sn , our results on |K(G, −∗ p∗ q ∗ )|
compare with the mass heuristic as follows. First, local masses µn,v∗ are given in
A DATABASE OF NUMBER FIELDS
15
the middle six columns:
(6.1)
n
5
6
7
∞
26/120
76/720
232/5040
2 3 5 7
40 19 27
145 83 31
180 99 55 57
tame
µ
7 5.31 .
11 6.39
15 5.18
The column µ contains the global mass 0.5µn,−∗ µn,p∗ µn,q∗ for two tame primes p
and q. When one or both of the primes are wild, the corresponding global mass is
substantially larger.
Tables 6.1, 6.2, and 6.3 clearly show that there tend to be more fields when
one or more of the primes p, q allows wild ramification, as one would expect from
(6.1). To make plausible conjectures about the asymptotic behavior of the numbers
|K(G, −∗ p∗ q ∗ )|, one would have to do more complicated local calculations than
those summarized in (6.1). These calculations would have to take into account
various secondary phenomena, such as the fact that s is forced to be even if p ≡
q ≡ 1 (4). Tables 6.1, 6.2, and 6.3 each reflect substantial computation, but the
amount of evidence is too small to warrant making formal conjectures in this setting.
7. Nilpotent octic fields with odd discriminant −s pa q b
The database has all octic fields with Galois group a 2-group and
discriminant
of the form −s pa q b with p and q odd primes < 250. There are 52
2 = 1326 pairs
{p, q} and the average size of K(NilOct, −∗ p∗ q ∗ ) in this range is about 12.01. In
comparison with the nonsolvable cases discussed in the previous two sections, there
is much greater regularity in this setting. We exhibit some of the greater regularity
and explain how it makes some of the abstract considerations of [BE11, BP00] more
concrete.
Twenty-six of the fifty octic groups have 2-power order. Table 7.1 presents the
nonzero cardinalities, so that e.g. |K(8T j, −∗ 5∗ 29∗ )| = 4, 2, 2 for j = 19, 20, 21.
The repeated proportion (2, 1, 1) for these groups and other similar patterns are
due to the sibling phenomenon discussed in Section 4. Only the sixteen 2-groups
generated by ≤ 2 elements actually occur. Columns s3 , #, ν, T , and s are all
explained later in this section.
The main phenomenon presented in Table 7.1 is that the multiplicities presented
are highly repetitious, with e.g. the multiplicities presented for (5, 29) occurring for
all together eleven pairs (p, q), as indicated in the # column. The repetition is even
greater than indicated by the table itself. Namely if (p1 , q1 ) and (p2 , q2 ) correspond
to the same line, then not only are the numbers K(8T j, −∗ p∗i qi∗ ) independent of i,
but the individual K(8T j, −∗ pai qib ) and even further refinements are also independent of i.
The line corresponding to a given pair (p, q) is almost determined by elementary
considerations as follows. Let U be the order of q in (Z/p)× and let V be the
order of p in (Z/q)× . Let u = gcd(U, 4) and v = gcd(V, 4). Then all (p, q) on a
given line have the same u, v, and a representative is written (pu , qv ) in the left
two columns. Almost all lines are determined by their datum {[p]u , [q]v }, with [·]
indicating reduction modulo 8. The only exceptions are {[p]u , [q]v } = {54 , 14 } and
{[p]u , [q]v } = {14 , 14 } which have two lines each. The column headed by # gives the
number of occurrences in our setting p, q < 250. In the five cases where this number
is less than 10 we continued the computation up through p, q < 500 assuming GRH.
16
JOHN W. JONES AND DAVID P. ROBERTS
p
q
1 2 4 5 6 7 8 10 16 17 19 20 21 27 28 30 s3
3, 7
72
112
3, 7
31
71
1
1
3, 7
31
114
192
5
51
52
52
1
1 2
1 2
3, 7
31
192
234
1
171
172
412
2 1
2 1 2
2 1 2
5
51
52
134
132
5
131
292
534
294
5
51
134
52
532
1094
1014
1
171
172
414
172
734
974
4
4
4
4
4
4
3
3
3
3
3
3
1
1
171 411
414 732
412 2412
734 894
734 1374
12
12
12
12
12
3
3
3
3
3
3
3
3
3
ν
4 193
4 185
1/4
1/8
1/8
4
4 219
6 87
7 86
1/4
1/8
1/16
1/16
4
4 162
6 66
8 52
1/4
1/8
1/16
1/16
1
2
1
2
1
2 1 1
4 1 2
3
2
3 3 1
3 3 1 2
2
3 3 1
3
3
3
3
3
#
1
2
1
2
1 2 2 2
1 4 2 4
1 4 2 4
1
6
1 4 6 4
1 6 6 6
1 6 6 6
2
2
2
2
6
6
6
6
6
6
6
6
6
6
6
6
2
2
4
2
2
4
2
2
8 4
8 12
2 12 4
2
8 4
4 4
12 4
2 16 12
2 16 12
2
4
2 16 4
6 24 12
6 24 12
1
1
2
1
1
2
6
2
5
2 4 4
9
6 12 12 16 11
2 2 2 4 9
2
2
2
6
6
5
9
9
9
11
12
1/8
76 1/16
17 1/64
22 1/64
18 1/64
6 1/128
1 1/128
6
9
4 10
16 12
24 13
1/16
27 1/32
12 1/64
2 1/128
2 1/256
2 1/256
2
2
6
6
4
2
4
2 4 4
2 2 2 4
2 2 2 4
6 12 12 16
6 16 16 24
2 4 4
2 4 4
6 16 16
6 24 24
T
s
◦
4
• ≥5
i
ii
iii
4
6
19
1/16
42 1/32
I
6
11 1/128 II
27
13 1/128 III ≥ 17
25 1/64 IV ≥ 30
Table 7.1. Nonzero cardinalities |K(8T j, −∗ p∗ q ∗ )| for 8T j an
octic group of 2-power order
We expect that all possibilities are accounted for by the table, and they occur
with asymptotic frequencies given in the column headed by ν. Assuming these
frequencies are correct, the average size of K(NilOct, −∗ p∗ q ∗ ) is exactly 15.875,
substantially larger than the observed 12.01 in the p, q < 250 setting.
The connection with [BE11, BP00] is as follows. Let L(p, q)k ⊂ C be the splitting
field of all degree 2k fields with Galois group a 2-group and discriminant −∗ p∗ q ∗ .
The Galois group Gal(L(p, q)k /Q) is a 2-group and so all ramification at the odd
primes p and q is tame. Let L(p, q) be the union of these L(p, q)k . The group
Gal(L(p, q)/Q) is a pro-2-group generated by the tame ramification elements τp
A DATABASE OF NUMBER FIELDS
17
and τq . The central question pursued in [BE11, BP00] is the distribution of the
Gal(L(p, q)/Q) as abstract groups.
Table 7.1 corresponds to working at the level of the quotient Gal(L(p, q)3 /Q).
The fact that this group has just the two generators τp and τq explains why only the
sixteen 2-groups having 1 or 2 generators appear. One has |Gal(L(p, q)3 /Q)| = 2s3
where s3 is as in Table 7.1. The lines with an entry under T are pursued theoretically
in [BE11]. The cases marked by ◦-•, i-iii, and I-IV are respectively treated in §5.2,
§5.3, and §5.4 there. The entire group Gal(L(p, q), Q) has order 2s , with s = ∞
being expected sometimes in Case IV .
Some of the behavior for k > 3 is previewed by 2-parts of class groups of octic fields. For example, in Case ii all 87 instances behave the same: the unique
fields in K(8T 2, −4 p3 q 7 ), K(8T 4, −4 p4 q 6 ), K(8T 17, −4 p6 q 5 ), and the two fields in
K(8T 17, −4 p6 q 7 ) all have 2 exactly dividing the class number; the remaining six
fields all have odd class number. In contrast, in Case iii the 86 instances break into
two types of behaviors, represented by (p, q) = (19, 5) and (p, q) = (11, 37). These
patterns on the database reflect the fact [BE11, §5.3] that in Case ii there is just
one possibility for (Gal(L(p, q)/Q); τp , τq ) while in Case iii there are two.
8. Nilpotent octic fields with discriminant −s 2a q b
The database has all octic fields with Galois group a 2-group and discriminant
of the form −s 2a q b with q < 2500. The sets K(NilOct, −∗ 2∗ q ∗ ) average 1711 fields,
the great increase from the previous section being due to the fact that now there
are many possibilities for wild ramification at 2. As in the previous section, there
is great regularity explained by identifications of relevant absolute Galois groups
[Koc02]. Again, even more so this time, there is further regularity not explained
by theoretical results.
Continuing with the notation of the previous section, consider the Galois extensions L(2, q) = ∪∞
k=1 L(2, q)k and their associated Galois groups Gal(L(2, q)/Q) =
lim Gal(L(2, q)k /Q). As before, octic fields with Galois group a 2-group let one
←−
study Gal(L(2, q)3 /Q). Table 8.1 presents summarizing data for q < 2500 in a format parallel to Table 7.1 but more condensed. Here the main entries count Galois
extensions of Q. Thus an entry m in the 192 20 21 column corresponds to m Galois
extensions of Q having degree 32. Each of these Galois extensions corresponds to
four fields on our database, of types 8T 19, 8T 19, 8T 20, and 8T 21.
In the range studied, there are thirteen different behaviors in terms of the cardinalities |K(8T j, −∗ 2∗ q ∗ )|. As indicated by Table 7.1, these cardinalities depend
mainly on the reduction of q modulo 16. However classes 1, 9, and 15 are broken into subclasses. The biggest subclasses have size |1A| = 23, |9A| = 24, and
|15A| = 28. The remaining subclasses are
1B = {113, 337, 353, 593, 881, 1249, 1777, 2113, 2129, 2273},
1C = {257, 1601},
1D = {577, 1201, 1217, 1553, 1889},
1E = {1153},
9B = {73, 281, 617, 1033, 1049, 1289, 1753, 1801, 1913, 2281, 2393},
9C = {137, 409, 809, 1129, 1321, 1657, 1993, 2137},
15B = {31, 191, 383, 607, 719, 863, 911, 991, 1103, 1231, 1327, 1471,
18
JOHN W. JONES AND DAVID P. ROBERTS
1487, 1567, 1583, 2063, 2111, 2287, 2351, 2383}.
A prime q ≡ 1 (16) is in 1A if and only if 2 6∈ F×4
q . Otherwise we do not have a
concise description of these decompositions.
|G| = 8
q
Q2
1A
1B
1C
1D
1E
9A
9B
9C
3, 11
5, 13
7
15A
15B
1
24
24
”
”
”
”
24
”
”
4
8
4
4
”
2
18
18
”
”
”
”
18
”
”
6
18
6
6
”
3
1
1
”
”
”
”
1
”
”
1
1
1
1
”
4
18
30
”
”
”
”
30
”
”
14
12
20
20
”
|G| = 16
5
6
2
”
”
”
”
2
”
”
2
0
0
0
”
62
16
42
54
”
”
”
44
”
”
10
10
30
32
”
7
36
36
36
”
”
”
36
”
”
6
20
6
6
”
8
36
44
60
”
”
”
48
”
”
22
10
16
16
”
94
9
15
”
”
”
”
15
”
”
7
6
10
10
”
102
12
36
”
”
”
”
36
”
”
4
12
12
12
”
|G| = 32
113
16
12
”
”
”
”
12
”
”
6
6
4
4
”
152
38
64
84
”
”
”
68
”
”
21
21
34
36
”
162
12
36
60
”
”
”
36
”
”
4
12
12
16
”
172 188
48 4
96 16
144 ”
” ”
” ”
” ”
112 16
” ”
” ”
8 3
36 1
24 8
24 8
” ”
21
20
192
24
48
96
”
”
”
48
”
”
4
12
12
20
”
|G| = 64
282 312
4
26 272 296 304
24 48 16 24
80 104 32 52
144 256 80 128
” 272 ” 136
” 336 ” 168
” ” ” ”
96 104 48 52
” ” ” ”
” 72 ” 36
16 8 8 4
6 24 4 2
44 24 20 12
52 64 24 32
” ” ” ”
358
48
72
312
336
384
240
156
132
156
21
9
60
96
84
Tot
1449
2895
6783
7071
7839
6687
3807
3615
3615
579
621
1401
2041
1945
Table 8.1. The q-j entry gives the number of Galois extensions
of Q with Galois group 8T j and discriminant of the form −s 2a q b .
The number of Galois extensions of Q2 with Galois group 8T j is
also given.
Let D∞ = {1, c} where c is complex conjugation. Let Dq ⊆ Gal(L(2, q)/Q) be a
q-decomposition group. Then, working always in the category of pro-2-groups, one
has the presentation Dq = hτ, σ|σ −1 τ σ = τ q i; here τ is a ramification element and
σ is a Frobenius element. Representing a more general theory, for q ≡ 3, 5 (8) one
has two remarkable facts [Koc02, Example 11.18]. First, the 2-decomposition group
D2 is all of Gal(L(2, q)/Q). Second, the global Galois group is a free product:
(8.1)
Gal(L(2, q)/Q) = D∞ ∗ Dq .
As a consequence, always for q ≡ 3, 5 (8), the quotients Gal(L(2, q)k /Q) are computable as abstract finite groups and moreover depend only on q modulo 8. In
particular, the counts on the lines 3,11 and 5,13 of Table 8.1 can be obtained
purely group-theoretically. The other lines of Table 8.1 are not covered by the
theory in [Koc02].
A important aspect of the situation is not understood theoretically, namely
the wild ramification at 2. The database exhibits extraordinary regularity at
the level k = 3 as follows. By 2-adically completing octic number fields K ∈
K(NilOct, −∗ 2∗ q ∗ ), one gets 579 octic 2-adic fields if q ≡ 3 (8) and 621 octic 2-adic
fields if q ≡ 5 (2). The regularity is that the subset of all 1499 nilpotent octic 2adic fields which arise depends on q only modulo 8, at least in our range q < 2500.
One can see some of this statement directly from the database: the cardinalities
|K(8T j, −∗ 2a q ∗ )| for given (j, a) depend only on q modulo 8.
A DATABASE OF NUMBER FIELDS
19
In the cases q ≡ 3, 5 (8), the group Gal(L(2, q)/Q) = D2 has a filtration by higher
ramification groups. From the group-theoretical description of Gal(L(2, q)/Q), one
can calculate that the quotient group Gal(L(2, q)3 /Q) has size 218 . The 18 slopes
measuring wildness of 2-adic ramification work out to
3 : 0,
5 : 0, 0,
2, 2, 2 12
2, 2, 2, 2 21
3, 3, 3 21 , 3 12 , 3 58 , 3 34 ,
3, 3, 3, 3 12 , 3 21 , 3 34 ,
4, 4, 4 41 , 4 14 , 4 83 , 4 12 , 4 34
4, 4 14 , 4 12 , 4 34 , 4 34 ,
5
5.
Most of these slopes can be read off from the octic field part of the database directly,
via the automatic 2-adic analysis of fields given there. For example, the first four
slopes for q = 3 all arise already from Q[x]/(x8 + 6x4 − 3), the unique member of
K(8T 8, −3 216 37 ). A few of the listed slopes can only be seen directly by working
with degree sixteen resolvents. A natural question, not addressed in the literature,
is to similarly describe the slopes appearing in all of Gal(L(2, q)/Q).
9. Minimal nonsolvable fields with grd ≤ Ω
Our focus for the remainder of the paper is on Galois number fields, for which
root discriminants and Galois root discriminants naturally coincide. As reviewed
in the introduction, in [JR07] we raised the problem of completely understanding
the set K[Ω] of all Galois number fields K ⊂ C with grd at most the Serre-Odlyzko
constant Ω = 8πeγ ≈ 44.76. As in [JR07], we focus attention here on the interesting
subproblem of identifying the subset Kns [Ω] of K which are nonsolvable. Our
last two sections explain how the database explicitly exhibits a substantial part of
Kns [Ω].
Β
Β
3
2
2
1
1
0
1
2
3
4
Α
0
1
2
3
4
Α
Figure 9.1.
Galois root discriminants 2α 3β (left) and 2α 5β
(right) arising from minimal nonsolvable fields of degree ≤ 11 in
the database. The lines have equation 2α q β = Ω.
We say a nonsolvable number field is minimal if it does not contain a strictly
smaller nonsolvable number field. So fields with Galois group say Sn are minimal,
while fields with Galois group say Cp × Sn or Cpk : Sn are not. Figure 9.1 draws a
ns
dot for each minimal nonsolvable field K1 ∈ Kmin
[Ω] coming the degree ≤ 11 part
20
JOHN W. JONES AND DAVID P. ROBERTS
of the database with grd of the form 2α 3β or 2α 5β . There are 654 fields in the first
case and 885 in the second. Of these fields, 24 and 17 have grd ≤ Ω. Figure 9.1
illustrates the extreme extent to which the low grd problem is focused on the least
ramified of all Galois number fields.
Figure 9.1 also provides some context for the next section as follows. Consider
the compositum K = K1 K2 of distinct minimal fields K1 and K2 contributing to
the same half of Figure 9.1. Let 2αi q βi be the root discriminant of Ki . The root
discriminant 2α q β of K satisfies α ≥ max(α1 , α2 ) and β ≥ max(β1 , β2 ). The figure
makes it clear that one must have almost exact agreement α1 ≈ α2 and β1 ≈ β2
for K to even have a chance of lying in Kns [Ω]. As some examples where one has
exact agreement, consider the respective splitting fields K1 , K2 , and K3 of
f1 (x)
= x5 − 10x3 − 20x2 + 110x + 116,
f2 (x)
= x5 + 10x3 − 10x2 + 35x − 18,
f3 (x)
= x5 + 10x3 − 40x2 + 60x − 32.
All three fields have Galois group A5 and root discriminant 23/2 58/5 ≈ 37.14. The
first two completely agree at 2, but differ at 5, so that K1 K2 has root discriminant
23/2 548/25 ≈ 62.17. The other two composita also have root discriminant well over
Ω, with grd(K2 K3 ) = 29/4 58/5 ≈ 62.47 and grd(K1 K3 ) = 29/4 548/25 ≈ 104.55.
These computations, done automatically by entering fi (x)fj (x) into the grd calculator of [JR06], are clear illustrations of the general difficulty of using known fields
in K[Ω] to obtain others.
ns
In [JR07], we listed fields proving |Kmin
[Ω]| ≥ 373. Presently, the fields on
ns
the database show |Kmin [Ω]| ≥ 386. In [JR07], we highlighted the fact that the
only simple groups involved were the five smallest, A5 , SL3 (2), A6 , SL2 (8) and
P SL2 (11), and the eighth, A7 . The new fields add SL2 (16) , G2 (2)0 , and A8 to the
list of simple groups involved. These groups are 10th , 12th , and tied for 19th on the
list of all non-abelian simple groups in increasing order of size.
#
|H|
1
60
2
168
3
360
4
504
5
660
8
2520
12
3600
10
4080
12
6048
19 20160
G=H
#
G = H.Q
#
A5
78
S5
192
SL3 (2)
.18
P GL2 (7)
...23
A6
5 S6 , P GL2 (9), M10 , P ΓL2 (9) 13, ....6 , 0 , .15
SL2 (8)
15
ΣL2 (8)
15
P SL2 (11)
1
P GL2 (11)
0
A7
1
S7
1
A25
A25 .2, A25 .V, A25 .C4 , A25 .D4
1 , .1 , 0 , 0
SL2 (16)
.1
SL2 (16).2, SL2 (16).4
0,0
G2 (2)0
0
G2 (2)
.1
A8
0
S8
.1
Table 9.1. Lower bounds on |K[G, Ω]| for minimal nonsolvable
groups G. Entries highlighted in bold are completeness results
from [JR07]. Fields found since [JR07] are indicated by .’s.
ns
Table 9.1 summarizes all fields on the database in Kmin
[Ω]. It is organized by
the socle H ⊆ G, which is a simple group except in the single case H = A5 × A5 .
The .’s indicate that, for example, of the 23 known fields in K[P GL2 (7), Ω], twenty
A DATABASE OF NUMBER FIELDS
21
are listed in [JR07] and three are new. The polynomial for the SL2 (16) field was
found by Bosman [Bos07], starting from a classical modular form of weight 2. We
found polynomials for the new SL3 (2) field and the three new P GL2 (7) fields
starting from Schaeffer’s list [Sch12, App A] of ethereal modular forms of weight 1.
Polynomials for the other new fields were found by specializing families. All fields
summarized by Table 9.1 come from the part of the database in degree ≤ 11, except
for Bosman’s degree seventeen polynomial and the degree twenty-eight polynomial
for G2 (2). It would be of interest to pursue calculations with modular forms more
ns
systematically. They have the potential not only to yield new fields in Kmin
[Ω], but
also to prove completeness for certain G.
10. General nonsolvable fields with grd ≤ Ω
We continue in the framework of the previous section, so that the focus remains
ns
on Galois number fields contained in C. For K1 ∈ Kmin
[Ω] such a Galois number
ns
field, let K[K1 ; Ω] be the subset of K [Ω] consisting of fields containing K1 . Clearly
[
(10.1)
Kns [Ω] =
K[K1 ; Ω].
K1
So a natural approach to studying all of Kns [Ω] is to study each K[K1 ; Ω] separately.
The refined local information contained in the database can be used to find
fields in K[K1 ; Ω]. The set of fields so obtained is always very small, often just
{K1 }. Usually it seems likely that the set of fields obtained is all of K[K1 ; Ω], and
sometimes this expectation is provable under GRH. We sketch such a proof for a
particular K1 in the first example below. In the remaining examples, we start from
other K1 and now construct proper extensions K ∈ K[K1 ; Ω], illustrating several
phenomena. Our examples are organized in terms of increasing degree [K : Q]. The
fields here are all extremely lightly ramified for their Galois group, and therefore
worthy of individual attention.
Our local analysis of a Galois number field K centers on the notion of p-adic
slope content described in [JR06, §3.4] and automated on the associated database.
Thus a p-adic slope content of [s1 , . . . , sm ]ut indicates a wild inertia group P of order
pm , a tame inertia group I/P of order t, and an unramified quotient D/I of order
u. Wild slopes si ∈ Q ∩ (1, ∞) are listed in weakly increasing order and from [JR06,
Eq. 7] the contribution pα to the root discriminant of K is determined by
!
m
X
1 t−1
p−1
sn+1−i + m
.
α=
i
p
p
t
i=1
The quantities t and u are omitted from presentations of slope content when they
are 1.
Degree 120 and nothing more from S5 . The polynomial
f1 (x) = x5 + x3 + x − 1
has splitting field K1 with root discriminant ∆1 = 112/3 371/2 ≈ 30.09. Since
∆1 22/3 ≈ 47.76, ∆1 31/2 ≈ 52.11, ∆1 111/6 ≈ 44.87, and ∆1 371/4 ≈ 74.20 are all
more than Ω, any K ∈ K[K1 ; Ω] has to have root discriminant ∆ = ∆1 . The GRH
bounds say that a field with root discriminant 30.09 can have degree at most 2400
[Mar82].
22
JOHN W. JONES AND DAVID P. ROBERTS
The main part of the argument is to use the database to show that most other a
priori possible G in fact do not arise as Gal(K/Q) for K ∈ K[K1 ; Ω]. For example,
if there were an S3 field K2 with absolute discriminant 112 37, then K1 K2 would
be in K[K1 ; Ω]; there is in fact an S3 field with absolute discriminant 11 · 372 , but
not one with absolute discriminant 112 37. As an example of a group that needs a
supplementary argument to be eliminated, the central extension G = 2.S5 does not
appear because the degree 12 subfield of K1 fixed by D5 ⊂ S5 has root discriminant
∆1 and class number 1.
Degree 1920 from A5 . The smallest root discriminant of any nonsolvable Galois
field is 26/7 172/3 ≈ 18.70 coming from a field K1 with Galois group A5 . This case
is complicated because one can add ramification in several incompatible directions,
so that there are different maximal fields in K[K1 ; Ω]. One overfield is the splitting
field K̃1 of f− (x) where
f± (x) = x10 + 2x6 ± 4x4 − 3x2 ± 4.
In this direction, ramification has been added at 2 making the slope content there
[2, 2, 2, 2, 4]6 and the root discriminant 239/16 172/3 ≈ 35.81. The only solvable field
K2 on the database which is not contained in K̃1 but has rd(K̃1 K2 ) < Ω is Q(i).
The field K̃1 K2 is the splitting field of f+ (x) with Galois group 10T 36. There is
yet another wild slope of 2, making the root discriminant 279/32 172/3 ≈ 36.60.
Degree 25080 from P SL2 (11). The only known field K1 with Galois group P SL2 (11)
and root discriminant less than Ω first appeared in [KM01] and is the splitting field
of
f1 (x) = x11 − 2x10 + 3x9 + 2x8 − 5x7 + 16x6 − 10x5 + 10x4 + 2x3 − 3x2 + 4x − 1.
The root discriminant is ∆1 = 18311/2 ≈ 42.79, forcing all members of K[K1 ; Ω] to
have root discriminant 18311/2 as well.
The prime 1831 is congruent to 3 modulo 4, so that the associated quadratic field
√
Q( −1831) is imaginary and its class number can be expected to be considerably
larger than 1. This class number is in fact 19, and the splitting field of a degree 19
polynomial in the database is the corresponding Hilbert class field K2 . The field
K1 K2 ∈ K[K1 ; Ω] has degree 660 · 38 = 25080.
Degree 48384 from SL2 (8).3. The splitting field K1 of
f1 (x) = x9 − 3x8 + 4x7 + 16x2 + 8x + 8
has Galois group Gal(K1 /Q) = 9T 32 = SL2 (8).3 and root discriminant 273/28 78/9 ≈
34.36. This root discriminant is the smallest known from a field with Galois group
SL2 (8).3. In fact, it is small enough that it is possible to add ramification at both
2 and 7 and still keep the root discriminant less than Ω. Namely let
f2 (x)
= x4 − 2x3 + 2x2 + 2,
f3 (x)
= x4 − x3 + 3x2 − 4x + 2.
The splitting fields K2 and K3 have Galois groups A4 and D4 respectively. Composing with K2 increases degree by four and adds wild slopes 2 and 2 to the original
2-adic slope content [20/7, 20/7, 20/7]37 . Composing with K3 then increases degrees
by 8, adding another wild slope of 2 to the 2-adic slope content and increasing
A DATABASE OF NUMBER FIELDS
23
the 7-adic tame degree from 9 to 36. The root discriminant of K1 K2 K3 is then
2153/56 735/36 ≈ 44.06.
Degree 80640 from S8 . The largest group in Table 9.1 is S8 , and the only known
field in K[S8 , Ω] is the splitting field K1 of
f1 (x) = x8 − 4x7 + 4x6 + 8x3 − 32x2 + 32x − 20.
Here Galois slope contents are [15/4, 7/2, 7/2, 3, 2, 2]3 and [ ]7 at 2 and 5 respectively, giving root discriminant 2111/32 56/7 ≈ 43.99. The only field on the database
which can be used to give a larger field in K[K1 ; Ω] is K2 = Q(i). This field gives an
extra wild slope of 2, raising the degree of K1 K2 to 80640 and the root discriminant
to 2223/64 56/7 ≈ 44.47.
Degree 86400 from A25 .V . Another new field K1 on Table 9.1, found by Driver, is
the splitting field of
f1 (x) = x10 − 2x9 + 5x8 − 10x6 + 28x5 − 26x4 − 5x2 + 50x − 25.
Like in the previous example, this field K1 is wildly ramified at 2 and tamely ramified at 5. Slope contents are [23/6, 23/6, 3, 8/3, 8/3]3 and [ ]6 for a root discriminant
of 2169/48 55/6 ≈ 43.89. The splitting field K2 of x3 − x2 + 2x + 2 has Galois group
S3 , with 2-adic slope content [3] and 5-adic slope content [ ]3 . In the compositum
K1 K2 , the extra slope is in fact 2 giving a root discriminant of 285/24 55/6 ≈ 44.53.
Degree 172800 from S5 and S6 . Consider the 386
= 74305 composita K1 K2 , as
2
ns
K1 and K2 vary over distinct known fields in Kmin
[Ω]. From our discussion of
Figure 9.1, one would expect that very few of these composita would have root
discriminant less than Ω. In fact, calculation shows that exactly one of these
composita has rd(K1 K2 ) ≤ Ω, namely the joint splitting field of
f1 (x)
=
x5 − x4 − x3 + 3x2 − x − 19,
f2 (x)
=
x6 − 2x5 + 4x4 − 8x3 + 2x2 + 24x − 20.
Here Gal(K1 /Q) = S5 and Gal(K2 /Q) = S6 . Both fields have tame ramification
of order 2 at 3 and order 5 at 7. Both are otherwise ramified only at 2, with K1
having slope content [2, 3]2 and K2 having slope content [2, 2, 3]3 . In the compositum K1 K2 , there is partial cancellation between the two wild slopes of 3, and the
slope content is [2, 2, 2, 2, 3]6 . The root discriminant of K1 K2 then works out to
239/16 31/2 74/5 ≈ 44.50. The existence of this remarkable compositum contradicts
Corollary 12.1 of [JR07] and is the only error we have found in [JR07].
2 9 1 4
The field discriminants of f1 and f2 are respectively −2 26 31 74 and −√
2 3 7 .
3) and
The
splitting
fields
K
and
K
thus
contain
distinct
quadratic
fields,
Q(
1
2
√
Q( 6) respectively. The compositum therefore has Galois group all of S5 × S6 , and
so the degree [K1 K2√: Q] = 120·720 = 86400 ties
√ that
√ of the previous example. But,
moreover, K3 = Q( −3) is disjoint from Q( 3, 6) and does not introduce more
ramification. So K = K1 K2 K3 has the same root discriminant 239/16 31/2 74/5 ≈
44.50, but the larger degree 2 · 86400 = 172800.
The GRH upper bound on degree for a given root discriminant δ ∈ [1, Ω) increases to infinity as δ increases to Ω (as illustrated by Figure 4.1 of [JR07]).
However, we have only exhibited fields K here of degree ≤ 172800. Dropping the
24
JOHN W. JONES AND DAVID P. ROBERTS
restriction that K is Galois and nonsolvable may let one obtain somewhat larger degrees, but there remains a substantial and intriguing gap between degrees of known
fields and analytic upper bounds on degree.
References
[BE11]
[Bel]
[Bel04]
[Bha05]
[Bha07]
[Bha10]
[BM83]
[BMO90]
[Bor]
[Bos07]
[BP00]
[Dah08]
[DJ10]
[GAP06]
[HKS06]
[Hoe09]
[Hun57]
[JRa]
[JRb]
[JR99]
[JR03]
[JR06]
[JR07]
Nigel Boston and Jordan S. Ellenberg, Random pro-p groups, braid groups, and random tame Galois groups, Groups Geom. Dyn. 5 (2011), no. 2, 265–280. MR 2782173
(2012b:11172)
Karim Belabas, cubic v. 1.2, http://www.math.u-bordeaux1.fr/~kbelabas/
research/software/cubic-1.2.tgz.
, On quadratic fields with large 3-rank, Math. Comp. 73 (2004), no. 248, 2061–
2074 (electronic). MR MR2059751 (2005c:11132)
Manjul Bhargava, The density of discriminants of quartic rings and fields, Ann. of
Math. (2) 162 (2005), no. 2, 1031–1063. MR 2183288 (2006m:11163)
, Mass formulae for extensions of local fields, and conjectures on the density
of number field discriminants, Int. Math. Res. Not. IMRN (2007), no. 17, Art. ID
rnm052, 20. MR 2354798 (2009e:11220)
, The density of discriminants of quintic rings and fields, Ann. of Math. (2)
172 (2010), no. 3, 1559–1591. MR 2745272 (2011k:11152)
Gregory Butler and John McKay, The transitive groups of degree up to eleven, Comm.
Algebra 11 (1983), no. 8, 863–911.
A.-M. Bergé, J. Martinet, and M. Olivier, The computation of sextic fields with a quadratic subfield, Math. Comp. 54 (1990), no. 190, 869–884. MR 1011438 (90k:11169)
Bordeaux tables of number fields, http://pari.math.u-bordeaux.fr/pub/pari/
packages/nftables/.
Johan Bosman, A polynomial with Galois group SL2 (F16 ), LMS J. Comput. Math.
10 (2007), 1461–1570 (electronic). MR 2365691 (2008k:12008)
Nigel Boston and David Perry, Maximal 2-extensions with restricted ramification, J.
Algebra 232 (2000), no. 2, 664–672. MR 1792749 (2001k:12005)
S. R. Dahmen, Classical and modular methods applied to diophantine equations, Ph.D.
thesis, University of Utrecht, 2008.
Eric D. Driver and John W. Jones, Minimum discriminants of imprimitive decic
fields, Experiment. Math. 19 (2010), no. 4, 475–479. MR 2778659 (2012a:11169)
The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4, 2006,
(http://www.gap-system.org).
Klaus Hulek, Remke Kloosterman, and Matthias Schütt, Modularity of Calabi-Yau
varieties, Global aspects of complex geometry, Springer, Berlin, 2006, pp. 271–309.
MR 2264114 (2007g:11052)
Jing Long Hoelscher, Infinite class field towers, Math. Ann. 344 (2009), no. 4, 923–
928. MR 2507631 (2010h:11185)
John Hunter, The minimum discriminants of quintic fields, Proc. Glasgow Math.
Assoc. 3 (1957), 57–67. MR 0091309 (19,944b)
John W. Jones and David P. Roberts, Artin L-functions with small conductor, in
preparation.
, The tame-wild principle for discriminant relations for number fields, to appear in Algebra and Number Theory.
, Sextic number fields with discriminant (−1)j 2a 3b , Number theory (Ottawa,
ON, 1996), CRM Proc. Lecture Notes, vol. 19, Amer. Math. Soc., Providence, RI,
1999, pp. 141–172. MR 2000b:11142
, Septic fields with discriminant ±2a 3b , Math. Comp. 72 (2003), no. 244,
1975–1985 (electronic). MR MR1986816 (2004e:11119)
, A database of local fields, J. Symbolic Comput. 41 (2006), no. 1, 80–97,
website: http://math.asu.edu/~jj/localfields.
, Galois number fields with small root discriminant, J. Number Theory 122
(2007), no. 2, 379–407. MR 2292261 (2008e:11140)
A DATABASE OF NUMBER FIELDS
25
[JR08]
, Number fields ramified at one prime, Algorithmic number theory, Lecture
Notes in Comput. Sci., vol. 5011, Springer, Berlin, 2008, pp. 226–239. MR 2467849
(2010b:11152)
[JW12]
John W. Jones and Rachel Wallington, Number fields with solvable Galois groups and
small Galois root discriminants, Math. Comp. 81 (2012), no. 277, 555–567.
[KM01]
Jürgen Klüners and Gunter Malle, A database for field extensions of the rationals,
LMS J. Comput. Math. 4 (2001), 182–196 (electronic). MR 2003i:11184
[Koc02]
Helmut Koch, Galois theory of p-extensions, Springer Monographs in Mathematics,
Springer-Verlag, Berlin, 2002, With a foreword by I. R. Shafarevich, Translated from
the 1970 German original by Franz Lemmermeyer, With a postscript by the author
and Lemmermeyer. MR 1930372 (2003f:11181)
[Mal02]
Gunter Malle, On the distribution of Galois groups, J. Number Theory 92 (2002),
no. 2, 315–329. MR 1884706 (2002k:12010)
[Mar82]
Jacques Martinet, Petits discriminants des corps de nombres, Number theory days,
1980 (Exeter, 1980), London Math. Soc. Lecture Note Ser., vol. 56, Cambridge Univ.
Press, Cambridge, 1982, pp. 151–193. MR 84g:12009
[MM99]
Gunter Malle and B. Heinrich Matzat, Inverse Galois theory, Springer Monographs
in Mathematics, Springer-Verlag, Berlin, 1999. MR 1711577 (2000k:12004)
[Odl90]
A. M. Odlyzko, Bounds for discriminants and related estimates for class numbers,
regulators and zeros of zeta functions: a survey of recent results, Sém. Théor. Nombres
Bordeaux (2) 2 (1990), no. 1, 119–141. MR MR1061762 (91i:11154)
[OT05]
Ken Ono and Yuichiro Taguchi, 2-adic properties of certain modular forms and their
applications to arithmetic functions, Int. J. Number Theory 1 (2005), no. 1, 75–101.
MR 2172333 (2006e:11057)
[PAR13]
The PARI Group, Bordeaux, Pari/gp, version 2.6.2, 2013.
[Poh82]
Michael Pohst, On the computation of number fields of small discriminants including
the minimum discriminants of sixth degree fields, J. Number Theory 14 (1982), no. 1,
99–117. MR 644904 (83g:12009)
[Rob]
David P. Roberts, Chebyshev covers and exceptional number vields, In preparation.
http://facultypages.morris.umn.edu/ roberts/.
[Rob07]
, Wild partitions and number theory, J. Integer Seq. 10 (2007), no. 6, Article
07.6.6, 34. MR 2335791 (2009b:11206)
[Sch12]
George Johann Schaeffer, The Hecke Stability Method and Ethereal Forms, ProQuest LLC, Ann Arbor, MI, 2012, Thesis (Ph.D.)–University of California, Berkeley.
MR 3093915
[SPDyD94] A. Schwarz, M. Pohst, and F. Diaz y Diaz, A table of quintic number fields, Math.
Comp. 63 (1994), no. 207, 361–376. MR 1219705 (94i:11108)
[Voi]
John Voight, Tables of totally real number fields, http://www.math.dartmouth.edu/
~jvoight/nf-tables/index.html.
[Voi08]
, Enumeration of totally real number fields of bounded root discriminant, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 5011, Springer, Berlin,
2008, pp. 268–281. MR 2467853 (2010a:11228)
School of Mathematical and Statistical Sciences, Arizona State University, PO Box
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Division of Science and Mathematics, University of Minnesota Morris, Morris, MN
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E-mail address: [email protected]